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Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = hypergeom([1/2, -n - 1, -n], [2, 2], 4*x).
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%I #14 Nov 22 2023 12:19:27

%S 1,1,1,1,3,2,1,6,12,5,1,10,40,50,14,1,15,100,250,210,42,1,21,210,875,

%T 1470,882,132,1,28,392,2450,6860,8232,3696,429,1,36,672,5880,24696,

%U 49392,44352,15444,1430,1,45,1080,12600,74088,222264,332640,231660,64350,4862

%N Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = hypergeom([1/2, -n - 1, -n], [2, 2], 4*x).

%F T(2*n, n) = Sum_{k=0..n} CatalanNumber(n)^2 * binomial(n + k, k).

%F From _Detlef Meya_, Nov 22 2023: (Start)

%F T(n, k) = binomial(n, k)*binomial(n+1, k)*binomial(2*k, k)/(k+1)^2.

%F T(n, k) = A001263(n+1, k+1)*binomial(2*k, k)/(k+1). (End)

%e Triangle T(n, k) starts:

%e [0] 1;

%e [1] 1, 1;

%e [2] 1, 3, 2;

%e [3] 1, 6, 12, 5;

%e [4] 1, 10, 40, 50, 14;

%e [5] 1, 15, 100, 250, 210, 42;

%e [6] 1, 21, 210, 875, 1470, 882, 132;

%e [7] 1, 28, 392, 2450, 6860, 8232, 3696, 429;

%e [8] 1, 36, 672, 5880, 24696, 49392, 44352, 15444, 1430;

%e [9] 1, 45, 1080, 12600, 74088, 222264, 332640, 231660, 64350, 4862;

%p p := n -> hypergeom([1/2, -n - 1, -n], [2, 2], 4*x):

%p T := (n, k) -> coeff(simplify(p(n)), x, k):

%p seq(seq(T(n, k), k = 0..n), n = 0..9);

%t T[n_,k_]:=Binomial[n,k]*Binomial[n+1,k]*Binomial[2*k,k]/(k+1)^2;Flatten[Table[T[n,k],{n,0,9},{k,0,n}]]

%t (* _Detlef Meya_, Nov 22 2023 *)

%Y Cf. A128088 (row sums), A358368 (central terms), A367022.

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Nov 06 2023